Phonon Modes of an Ionic Crystal Slab

Abstract

The vibrational problem of a slab‐shape diatomic ionic crystal is studied within the harmonic approximation and neglecting retardation effects. The interest is centered on a proper treatment of the physical boundary conditions existing at the free surfaces of the slab. By taking advantage of the translational symmetry of the slab for lattice translations parallel to the free surfaces, the problem is first reduced to finding the normal modes of a linear model consisting of two parallel diatomic chains coupled together. Using the so‐called planewise summation method developed by deWette and Schacher for computing the Coulomb interaction coefficients, it is shown that the consideration of nearest and n ext‐nearest‐neighbor interactions between the ions of this double‐chain problem is largely sufficient to determine the eigenmodes accurately. This remarkable feature arises from the fact that the Coulomb field created by a plane array of dipoles decreases exponentially with increasing distance from the plane. The double‐chain dynamics is then developed following the general method recently given by Gazis and Wallis. In this paper, we restrict the computation to this class of phonons which propagate perpendicularly to the boundary planes. We found that (i) surface modes of vibrations (with exponentially decreasing amplitude from the surfaces) exist, resulting from the use of physical boundary conditions; (ii) when the slab thickness increases, the “bulk” modes (with a wavelike character) rapidly converge to the solutions obtained with cyclic boundary conditions. It is concluded that, for sufficiently thick ionic slab‐shape crystals (a few tens of layers), the use of periodic boundary conditions does not significantly affect the statistical properties of the crystal (as shown by Ledermann's theorem in the case of short‐range force model) and also accurately provides the true individual eigenfrequencies with the exception, however, of this class of modes involving localized vibrations at the surfaces of the slab.